Finite Group Databasebeta

← C2×#20244697 #26244205

Subgroups — C2×#20244697 #26244205

Order 12032.

1918 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#10625334C211yesmaximal, characteristic
2C22×Dic47⋊C8C211yesmaximal, characteristic, complemented
3C22×#13093946C211yesmaximal, characteristic, complemented
4C2×#15689026C211yesmaximal, characteristic, complemented
5C2×#15689024C211yesmaximal, characteristic, complemented
6#20244697C218yesmaximal, direct product factor
7C2×#15688885C211yesmaximal, characteristic, complemented
8C22×#102×C47C211yesmaximal, characteristic, complemented
9C23×D188C2211yescharacteristic
10C22×#10625334—24no—
11C22×#10625334—24no—
12C22×#10625334—24no—
13C22×#10625334C2214yes—
14C23×C47⋊C8C2211yescharacteristic
15C22×C4⋊Dic47C2211yescharacteristic
16C22×C22⋊Dic47—21no—
17C22×C22⋊Dic47—21no—
18C22×(C2×D47)⋊C4—21no—
19C22×(C2×D47)⋊C4—21no—
20C22×C4×Dic47C2211yescharacteristic
21C2×Dic47⋊C8—24no—
22C2×Dic47⋊C8C2214yescomplemented
23C2×#13093946—24no—
24C2×#13093946C2214yescomplemented
25#15689026—22no—
26#15689026C2214yescomplemented
27#15689024—22no—
28#15689024C2214yescomplemented
29#15688885C2214yescomplemented
30C23×D4×C47C2211yescharacteristic
31C23×C8×C47C2211yescharacteristic
32C22×C4⋊C4×C47C2211yescharacteristic
33C2×#102×C47—24no—
34C2×#102×C47C2214yescomplemented
35C24×D47—21no—
36C22×D188—22no—
37C22×D188—22no—
38C22×D188—22no—
39C22×D188—22no—
40C22×D188—21no—
41C22×D188—22no—
42C22×D188D414yes—
43C22×D188C2312yes—
44C23×Dic47—21no—
45C23×Dic47D411yescharacteristic
46C23×Dic47D411yescharacteristic
47C2×#10625334—232no—
48C2×#10625334—28no—
49C2×#10625334—28no—
50C2×#10625334—28no—
51C22×C47⋊C8—22no—
52C22×C47⋊C8—22no—
53C22×C47⋊C8D414yes—
54C22×C47⋊C8C2312yes—
55C2×C4⋊Dic47—22no—
56C2×C4⋊Dic47—22no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47C2312yes—
59C2×C22⋊Dic47—24no—
60C2×C22⋊Dic47—24no—
61C2×C22⋊Dic47—44no—
62C2×C22⋊Dic47—44no—
63C2×(C2×D47)⋊C4—24no—
64C2×(C2×D47)⋊C4—24no—
65C2×(C2×D47)⋊C4—44no—
66C2×(C2×D47)⋊C4—44no—
67C2×C4×Dic47—22no—
68C2×C4×Dic47—22no—
69C2×C4×Dic47D412yes—
70C2×C4×Dic47C2312yes—
71Dic47⋊C8—28no—
72#13093946—28no—
73C25×C47—21no—
74C22×D4×C47—22no—
75C22×D4×C47—22no—
76C22×D4×C47—21no—
77C22×D4×C47—22no—
78C22×D4×C47—22no—
79C22×D4×C47—22no—
80C22×D4×C47D414yes—
81C22×D4×C47C2312yes—
82C23×C4×C47—21no—
83C23×C4×C47C2311yescharacteristic
84C22×C8×C47—22no—
85C22×C8×C47—22no—
86C22×C8×C47—22no—
87C22×C8×C47C2312yes—
88C2×C4⋊C4×C47—22no—
89C2×C4⋊C4×C47—22no—
90C2×C4⋊C4×C47—21no—
91C2×C4⋊C4×C47C2312yes—
92#102×C47—28no—
93C24×C47—24no—
94C24×C47—22no—
95C24×C47—42no—
96C24×C47—42no—
97C24×C47—42no—
98C24×C47—42no—
99C24×C47—42no—
100C24×C47—42no—
101C24×C47C2×D411yescharacteristic
102C23×D47—24no—
103C23×D47—22no—
104C23×D47—42no—
105C23×D47—42no—
106C23×D47—42no—
107C23×D47—42no—
108C23×D47—42no—
109C23×D47—42no—
110C2×D4×C47—28no—
111C2×D4×C47—24no—
112C2×D4×C47—28no—
113C2×D4×C47—22no—
114C2×D4×C47—22no—
115C2×D4×C47—22no—
116C2×D4×C47—22no—
117C2×D4×C47—22no—
118C2×D4×C47—42no—
119C2×D4×C47—42no—
120C2×D4×C47—44no—
121C2×D4×C47—44no—
122C2×D4×C47C2×D414yes—
123C22×C4×C47—22no—
124C22×C4×C47—42no—
125C22×C4×C47—42no—
126C22×C4×C47—42no—
127C22×C4×C47C2×D412yes—
128C22×C4×C47C2×D412yes—
129C22×C4×C47C2×D412yes—
130C22×C4×C47C2×D411yescharacteristic
131C22×C4×C47#4112yes—
132C22×C4×C47#4112yes—
133C22×C4×C47#4112yes—
134C22×C4×C47C2411yescharacteristic
135C22×Dic47—22no—
136C22×Dic47—22no—
137C22×Dic47—22no—
138C22×Dic47—22no—
139C22×Dic47—22no—
140C22×Dic47—22no—
141C22×Dic47—42no—
142C22×Dic47—42no—
143C22×Dic47—42no—
144C22×Dic47D814yescomplemented
145C22×Dic47C2×D412yes—
146C22×Dic47C2×D412yes—
147C2×D188—28no—
148C2×D188—28no—
149C2×D188—22no—
150C2×D188—22no—
151C2×D188—22no—
152C2×D188—22no—
153C2×D188—24no—
154C2×D188—22no—
155C2×D188—42no—
156C2×D188—42no—
157C2×D188—44no—
158C2×D188—44no—
159C2×D188C2×D414yes—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C2×C8×C47—44no—
164C4⋊C4×C47—22no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—42no—
167C22⋊Dic47—48no—
168C22⋊Dic47—48no—
169#10625334—232no—
170(C2×D47)⋊C4—48no—
171(C2×D47)⋊C4—48no—
172C4⋊Dic47—22no—
173C4⋊Dic47—22no—
174C4⋊Dic47—42no—
175C4×Dic47—24no—
176C4×Dic47—22no—
177C4×Dic47—22no—
178C2×C47⋊C8—28no—
179C2×C47⋊C8—22no—
180C2×C47⋊C8—22no—
181C2×C47⋊C8C2×D414yes—
182C23×C47—24no—
183C23×C47—42no—
184C23×C47—42no—
185C23×C47—42no—
186C23×C47—42no—
187C23×C47—42no—
188C23×C47—42no—
189C23×C47—48no—
190C23×C47—48no—
191C23×C47—48no—
192C23×C47—84no—
193C23×C47—84no—
194C23×C47—84no—
195C23×C47—84no—
196C23×C47#12112yes—
197C23×C47#12112yes—
198C23×C47#12112yes—
199C23×C47C2×D812yes—
200C23×C47#13512yes—
201C23×C47AGL(1,ℤ/8)12yes—
202C23×C47AGL(1,ℤ/8)12yes—
203C23×C47C22×D411yescharacteristic
204C22×D47—24no—
205C22×D47—42no—
206C22×D47—42no—
207C22×D47—42no—
208C22×D47—42no—
209C22×D47—42no—
210C22×D47—42no—
211C22×D47—48no—
212C22×D47—48no—
213C22×D47—48no—
214C22×D47—84no—
215C22×D47—84no—
216C22×D47—84no—
217C22×D47—84no—
218D4×C47—24no—
219D4×C47—24no—
220D4×C47—42no—
221D4×C47—42no—
222D4×C47—48no—
223C2×C4×C47—42no—
224C2×C4×C47—42no—
225C2×C4×C47—42no—
226C2×C4×C47—84no—
227C2×C4×C47#12014yes—
228C2×C4×C47#12112yes—
229C2×C4×C47#12112yes—
230C2×C4×C47#12112yes—
231C2×C4×C47#12314yes—
232C2×C4×C47#12314yes—
233C2×C4×C47#12414yes—
234C2×C4×C47C22×D411yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C22×D411yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47C2×#4111yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×Dic47—22no—
241C2×Dic47—22no—
242C2×Dic47—28no—
243C2×Dic47—22no—
244C2×Dic47—22no—
245C2×Dic47—22no—
246C2×Dic47—44no—
247C2×Dic47—42no—
248C2×Dic47—42no—
249C2×Dic47—42no—
250C2×Dic47—84no—
251C2×Dic47C2×D814yescomplemented
252D188—24no—
253D188—24no—
254D188—42no—
255D188—42no—
256D188—48no—
257C8×C47—42no—
258C47⋊C8—24no—
259C2×#2054—471nomaximal, Sylow
260C22×C47—44no—
261C22×C47—44no—
262C22×C47—44no—
263C22×C47—816no—
264C22×C47—82no—
265C22×C47—82no—
266C22×C47—82no—
267C22×C47—82no—
268C22×C47#39414yes—
269C22×C47#45914yes—
270C22×C47#45914yes—
271C22×C47#46314yes—
272C22×C47#46314yes—
273C22×C47#46614yes—
274C22×C47#46914yes—
275C22×C47C2×#12111yescharacteristic
276C22×C47C2×#12111yescharacteristic
277C22×C47C2×#12111yescharacteristic
278C22×C47C22×D811yescharacteristic
279C22×C47C2×#13511yescharacteristic
280C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
281C22×C47C2×AGL(1,ℤ/8)11yescharacteristic
282C2×D47—44no—
283C2×D47—44no—
284C2×D47—44no—
285C2×D47—816no—
286C2×D47—82no—
287C2×D47—82no—
288C2×D47—82no—
289C2×D47—82no—
290C4×C47—82no—
291C4×C47#39414yes—
292C4×C47C2×#12011yescharacteristic
293C4×C47C2×#12311yescharacteristic
294C4×C47C2×#12311yescharacteristic
295C4×C47C2×#12411yescharacteristic
296Dic47—24no—
297Dic47—42no—
298Dic47—82no—
299C23×D8—471no—
300C22×C4⋊C8—471no—
301C22×#102—471no—
302C22×#102D4711yescharacteristic, complemented
303C2×#390—471no—
304C2×#390—471no—
305#2054—478no—
306C2×#340—471no—
307C2×C47—84no—
308C2×C47#205418yescomplemented
309C2×C47C2×#39411yescharacteristic
310C2×C47C2×#45911yescharacteristic
311C2×C47C2×#45911yescharacteristic
312C2×C47C2×#46311yescharacteristic
313C2×C47C2×#46311yescharacteristic
314C2×C47C2×#46611yescharacteristic
315C2×C47C2×#46911yescharacteristic
316D47—84no—
317C23×D4—471no—
318C23×D4C2×D4711yescharacteristic
319C22×D8—474no—
320C22×D8—944no—
321C22×D8—944no—
322C22×D8—944no—
323C23×C8—471no—
324C23×C8C2×D4711yescharacteristic
325C22×C4⋊C4—471no—
326C22×C4⋊C4C2×D4711yescharacteristic
327C22×C22⋊C4—941no—
328C22×C22⋊C4—941no—
329C22×C22⋊C4—941no—
330C22×C22⋊C4—941no—
331C22×C42—471no—
332C2×C4⋊C8—474no—
333C2×C4⋊C8—944no—
334C2×#102—24no—
335C2×#102—474no—
336C2×#102—944no—
337C2×#102C2×D4714yescomplemented
338#390—474no—
339#390—474no—
340#390—942no—
341#390—942no—
342#340—474no—
343C47C2×#205411yescharacteristic, complemented, Sylow
344C25—21no—
345C25—941no—
346C22×D4—22no—
347C22×D4—22no—
348C22×D4—22no—
349C22×D4—22no—
350C22×D4—21no—
351C22×D4—22no—
352C22×D4—474no—
353C22×D4—472no—
354C22×D4—942no—
355C22×D4—942no—
356C22×D4—942no—
357C22×D4—941no—
358C22×D4—942no—
359C22×D4—942no—
360C22×D4C22⋊D4714yes—
361C22×D4C22×D4712yes—
362C23×C4—21no—
363C23×C4—471no—
364C23×C4—471no—
365C23×C4—941no—
366C23×C4C22×D4711yescharacteristic
367C2×D8—948no—
368C2×D8—948no—
369C2×D8—9432no—
370C2×D8—948no—
371C22×C8—22no—
372C22×C8—22no—
373C22×C8—22no—
374C22×C8—474no—
375C22×C8—472no—
376C22×C8—942no—
377C22×C8—942no—
378C22×C8C22×D4712yes—
379C2×C4⋊C4—22no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—472no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—941no—
386C2×C4⋊C4C22×D4712yes—
387C2×C22⋊C4—944no—
388C2×C22⋊C4—944no—
389C2×C22⋊C4—944no—
390C2×C22⋊C4—944no—
391C2×C22⋊C4—1884no—
392C2×C22⋊C4—1884no—
393C2×C22⋊C4—1884no—
394C2×C22⋊C4—1884no—
395C2×C42—472no—
396C2×C42—472no—
397C2×C42—942no—
398C2×C42—942no—
399C4⋊C8—948no—
400#102—28no—
401#102—948no—
402C24—24no—
403C24—22no—
404C24—42no—
405C24—42no—
406C24—42no—
407C24—42no—
408C24—42no—
409C24—42no—
410C24—944no—
411C24—942no—
412C24—1882no—
413C24—1882no—
414C24—1882no—
415C24—1882no—
416C24—1882no—
417C24—1882no—
418C24D4×D4711yescharacteristic
419C2×D4—28no—
420C2×D4—24no—
421C2×D4—28no—
422C2×D4—22no—
423C2×D4—22no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—22no—
427C2×D4—42no—
428C2×D4—42no—
429C2×D4—44no—
430C2×D4—44no—
431C2×D4—474no—
432C2×D4—948no—
433C2×D4—944no—
434C2×D4—948no—
435C2×D4—942no—
436C2×D4—942no—
437C2×D4—942no—
438C2×D4—942no—
439C2×D4—942no—
440C2×D4—1882no—
441C2×D4—1882no—
442C2×D4—1884no—
443C2×D4—1884no—
444C2×D4C2×C22⋊D4714yes—
445C22×C4—22no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—474no—
450C22×C4—472no—
451C22×C4—472no—
452C22×C4—942no—
453C22×C4—942no—
454C22×C4—942no—
455C22×C4—942no—
456C22×C4—942no—
457C22×C4—942no—
458C22×C4—1882no—
459C22×C4—1882no—
460C22×C4—1882no—
461C22×C4#1062535012yes—
462C22×C4#1062535012yes—
463C22×C4D4×D4712yes—
464C22×C4D4×D4712yes—
465C22×C4D4×D4711yescharacteristic
466C22×C4#1062535212yes—
467C22×C4C2×C22⋊D4712yes—
468C22×C4C23×D4711yescharacteristic
469D8—9432no—
470C2×C8—22no—
471C2×C8—22no—
472C2×C8—22no—
473C2×C8—44no—
474C2×C8—474no—
475C2×C8—948no—
476C2×C8—942no—
477C2×C8—942no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—42no—
481C4⋊C4—942no—
482C4⋊C4—942no—
483C4⋊C4—1882no—
484C22⋊C4—1888no—
485C22⋊C4—1888no—
486C22⋊C4—1888no—
487C22⋊C4—1888no—
488C42—944no—
489C42—942no—
490C42—942no—
491C23—24no—
492C23—48no—
493C23—48no—
494C23—48no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—84no—
502C23—84no—
503C23—84no—
504C23—84no—
505C23—944no—
506C23—3764no—
507C23—3764no—
508C23—3764no—
509C23—3764no—
510C23—1888no—
511C23—1888no—
512C23—1888no—
513C23—1882no—
514C23—1882no—
515C23—1882no—
516C23—1882no—
517C23—1882no—
518C23—1882no—
519C23#1309399612yes—
520C23#1309399712yes—
521C23#1309399712yes—
522C23D8×D4712yes—
523C23#1309404512yes—
524C23#1309404812yes—
525C23#1309405112yes—
526C23C2×D4×D4711yescharacteristic
527D4—24no—
528D4—24no—
529D4—48no—
530D4—42no—
531D4—42no—
532D4—944no—
533D4—944no—
534D4—1888no—
535D4—1882no—
536D4—1882no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—84no—
541C2×C4—474no—
542C2×C4—948no—
543C2×C4—942no—
544C2×C4—942no—
545C2×C4—942no—
546C2×C4—942no—
547C2×C4—942no—
548C2×C4—3764no—
549C2×C4—1882no—
550C2×C4—1882no—
551C2×C4—1882no—
552C2×C4—1884no—
553C2×C4#1309399214yes—
554C2×C4#1309399612yes—
555C2×C4#1309399712yes—
556C2×C4#1309399712yes—
557C2×C4#1309400614yes—
558C2×C4#1309400714yes—
559C2×C4#1309401014yes—
560C2×C4C2×D4×D4711yescharacteristic
561C2×C4C2×D4×D4711yescharacteristic
562C2×C4C22×C22⋊D4711yescharacteristic
563C2×C4C2×#1062535211yescharacteristic
564C2×C4C2×#1062535011yescharacteristic
565C2×C4C2×#1062535011yescharacteristic
566C8—42no—
567C8—944no—
568C22—44no—
569C22—44no—
570C22—44no—
571C22—82no—
572C22—82no—
573C22—82no—
574C22—82no—
575C22—816no—
576C22—3762no—
577C22—3762no—
578C22—3762no—
579C22—3762no—
580C22—37616no—
581C22—1884no—
582C22—1884no—
583C22—1884no—
584C22#1568903814yes—
585C22#1568931614yes—
586C22#1568931714yes—
587C22#1568933714yes—
588C22#1568934014yes—
589C22#1568935814yes—
590C22#1568937514yes—
591C22C2×#1309399611yescharacteristic
592C22C2×#1309399711yescharacteristic
593C22C2×#1309399711yescharacteristic
594C22C2×D8×D4711yescharacteristic
595C22C2×#1309404511yescharacteristic
596C22C2×#1309404811yescharacteristic
597C22C2×#1309405111yescharacteristic
598C4—82no—
599C4—944no—
600C4—3762no—
601C4—1882no—
602C4#1568903814yes—
603C4C2×#1309399211yescharacteristic
604C4C2×#1309400611yescharacteristic
605C4C2×#1309400711yescharacteristic
606C4C2×#1309401011yescharacteristic
607C2—84no—
608C2—3764no—
609C2#2024469718yesdirect product factor
610C2C2×#1568903811yescharacteristic
611C2C2×#1568931611yescharacteristic
612C2C2×#1568931711yescharacteristic
613C2C2×#1568933711yescharacteristic
614C2C2×#1568934011yescharacteristic
615C2C2×#1568935811yescharacteristic
616C2C2×#1568937511yescharacteristic
subgroupG/H
centerC24D4×D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC22×#102×C47C2
socleC24×C47C2×D4
prime radicalC23×#10625334C2
hypercenterC22×#102D47
hypocenterC47C2×#2054
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C22×Dic47⋊C8C2semidirect
3. C22×#13093946C2semidirect
4. C2×#15689026C2semidirect
5. C2×#15689024C2semidirect
6. #20244697C2direct (N × H)
7. C2×#15688885C2semidirect
8. C22×#102×C47C2semidirect
22. C2×Dic47⋊C8C22semidirect
24. C2×#13093946C22semidirect
26. #15689026C22semidirect
28. #15689024C22semidirect
29. #15688885C22semidirect
34. C2×#102×C47C22semidirect
144. C22×Dic47D8semidirect
251. C2×Dic47C2×D8semidirect
302. C22×#102D47semidirect
308. C2×C47#2054semidirect
337. C2×#102C2×D47semidirect
343. C47C2×#2054semidirect
609. C2#20244697direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C23×#106253342
C22×Dic47⋊C82
C22×#130939462
C2×#156890262
C2×#156890242
#202446972
C2×#156888852
C22×#102×C472
C2×#205447